1. Technical Field of the Invention
The invention relates to determining the maximal Lyapunov exponent in a chaotic system. More particularly, the invention relates to determining the maximal Lyapunov exponent by comparing values calculated at different precision levels.
2. Description of the Related Art
Chaotic, non-linear systems are known. Examples of chaotic systems include, but are not limited to, the weather, the stock market and fluid flow. By its nature it is difficult to calculate the outcome of a chaotic system. A slight change in initial conditions or an intervening event can result in a completely different system trajectory and a completely different outcome. In a weather system, a slight change in the initial conditions of temperatures and pressures at several locations produces a different weather trajectory. As a result, a long range forecast may be difficult or impossible to make. One method of predicting an outcome of a chaotic system is by calculating a Lyapunov exponent. The Lyapunov exponent is a measure of the exponential rate of divergence of a system trajectory with respect to time.
It is well known that the significant digits of variables defining a chaotic system are rapidly lost during calculation and the initially valid digits in the chaotic system become meaningless in a small number of iterations. The number of valid digits n(t) decreases linearly with time. The maximal Lyapunov exponent L is equal to −dn/dt and can be calculated to determine the number of digits lost per unit time. Traditional Lyapunov exponent algorithms, such as those calculated by the Benettin technique, can be applied only to a system of differential equations. A different algorithm is used for discrete maps and no algorithm existed here-to-fore for any other mathematical methods.